Dispersion-less Kerr solitons in spectrally confined optical cavities

Solitons are self-reinforcing localized wave packets that manifest in the major areas of nonlinear science, from optics to biology and Bose–Einstein condensates. Recently, optically driven dissipative solitons have attracted great attention for the implementation of the chip-scale frequency combs that are decisive for communications, spectroscopy, neural computing, and quantum information processing. In the current understanding, the generation of temporal solitons involves the chromatic dispersion as a key enabling physical effect, acting either globally or locally on the cavity dynamics in a decisive way. Here, we report on a novel class of solitons, both theoretically and experimentally, which builds up in spectrally confined optical cavities when dispersion is practically absent, both globally and locally. Precisely, the interplay between the Kerr nonlinearity and spectral filtering results in an infinite hierarchy of eigenfunctions which, combined with optical gain, allow for the generation of stable dispersion-less dissipative solitons in a previously unexplored regime. When the filter order tends to infinity, we find an unexpected link between dissipative and conservative solitons, in the form of Nyquist-pulse-like solitons endowed with an ultra-flat spectrum. In contrast to the conventional dispersion-enabled nonlinear Schrödinger solitons, these dispersion-less Nyquist solitons build on a fully confined spectrum and their energy scaling is not constrained by the pulse duration. Dispersion-less soliton molecules and their deterministic transitioning to single solitons are also evidenced. These findings broaden the fundamental scope of the dissipative soliton paradigm and open new avenues for generating soliton pulses and frequency combs endowed with unprecedented temporal and spectral features.


Mean-field equation for dispersion-less cavity solitons
When only the n-th order spectral loss is considered, the field evolution can be described by the following mean-field equation S1-S3 where A is the field envelop; t is the time; z is the propagation distance;  is the uniform loss;  is the pump-cavity phase detuning;  is the Kerr nonlinear coefficient;  is the spectral loss coefficient; p A is the pump field; and  is the pump coupling ratio. Here we only consider the case when n is even. The spectral loss per unit length For bright solitons, C is the solution on the lower branch of the bi-stability curve S4 .

Eigen functions of spectral filtering and self-phase modulation 1.2.1 Evolution with filter order
The eigenfunctions of combined spectral filtering and self-phase modulation where  and  represent the real and imaginary parts of the eigenvalue respectively. It is noted that  and  are correlated. For a given  , the eigenfunction e U and the real eigenvalue  can be obtained by numerically solving Eq. (S7) with the Newton-Rapson method. Figure S1 shows the evolution of e U and  with the filter order when 1  = . It is found that  is always negative, representing an overall amplitude loss induced by SPM in combination with spectral filtering. Remarkably,  decreases with the increase of filter order n . When the filter order n →, we have 0  → .
Although the eigenfunctions do not necessarily constitute stable solitons, they provide important insights for the dispersion-less dissipative solitons (as is shown by the comparison in Fig. 1b of the main paper) and may be regarded as a kind of soliton "kernel". One particularly interesting case is when 0  → with n →. Note that the spectral loss can be expressed in the frequency domain as where  is the optical frequency. Therefore, when n →, it turns to an ideal gate filter with a unit bandwidth The vanishment of  implies that in this limiting case, an energy-conserved balance can be achieved between SPM and the gate bandpass filtering. The resulting waveform is close to a Nyquist pulse with a fully confined spectrum.

Scaling law
The scaling law for the eigenfunctions can be easily checked as where  is a positive number. The pulse energy is then where 0 E is the energy of ( ) e UT, given by And the pulse width is 1 0 where 0 W is the pulse width of ( ) e UT. The relation between E and W is thus given by 11 1 00 where 1 00 n n C E W − = is a constant depending on the filter order. The pulse energy scales with the ( ) 1 n − -th power of the inverse pulse duration. Again, the interesting condition is when n → . The pulse width will be nearly unchanged when the pulse energy increases (also see Eq. (S10)), as a result of a spectrum that is fully confined by an ideal gate filter.

Nyquist-pulse-like solution
Equation (S7) where H is the ideal gate filter (i.e., Eq. (S9)). The values of a and b may be found analytically by using the variational method. A much simpler and likely more accurate approach is just fitting the numerical results with Eq. (S15  Figure S2 compares the numerical results with the calculated results based on Eqs. (S15) and (S17), showing very good agreement. According to the scaling law given by Eq.
(S10), the eigenfunction spectrum for arbitrary  can be written as (S18) The results above are obtained with the normalized equation. For any practical experimental configuration, the soliton parameters can be easily obtained by performing denormalization.

Nyquist cavity solitons
In the next, based on Eq. (S15), we derive an approximate solution for the Nyquistpulse-like solitons in coherently driven cavities. The procedure is similar to perturbation analysis for conventional dispersion-driven solitons S4 , but performed in the frequency domain because the time-domain pulse shape of Eq. (S17) is complicated.
We first transform Eq. (S6) to the frequency domain as follows Suppose the soliton spectrum can be written as where  and  are spectral amplitude and phase respectively. For convenience, we use the simplified denotation and noticing that we will have The terms enclosed in square brackets represent parametric mixing between the soliton and the homogenous background. Under the small-perturbation assumption, one balance is achieved between SPM and gate spectral filtering, while the other is achieved between parametric gain and uniform loss. The frequency-dependence of parametric gain can then be neglected in approximate analysis. Thus, we assume Q Q kQ  where the constant 0.687 k = can be retrieved with numerical curve fitting. By separating the real and imagery terms of Eq. (S22), we obtain the following coupled When the phase detuning 1  , the lower-branch homogenous solution i CS  −  . Further neglecting the higher-order terms with 2 C , we get A comparison between the numerical results and the approximate analytical results is shown in Fig. S3, showing relatively good agreement. The parameters are 30 = and 2 60 S = . external pump will get the maximum gain when it is coherently combined with the pump field. It is noted that the analytical approximation by Eq. (S25) is more accurate with lower pump intensity under the premise of soliton sustainment. With the increase of pump power, the cavity soliton will show slightly increasing distortions in comparison to the eigenfunction, as is indicated by Fig. 4e of the main paper.
The soliton stability can be investigated according to Eq. (24). Figure S4 shows one simulated example in which the initial soliton parameters deviate from the stationary solution. With the increase of propagation distance, the soliton finally converges to the fixed point.

Nyquist soliton molecules
Depending the initial field (e.g., when the pulse duration is much wider than the transform-limited value), stable soliton molecules composed of multiple closely bound pulses can also be observed in simulations. These Nyquist soliton molecules are also related to the eigenfunctions of combined SPM and gate spectral filtering. Figure S5 shows the numerical results simulated based on Eqs.

Pulse pumped equation for the fiber cavities
When the Kerr cavity is pumped by optical pulses, a new term should be introduced to account for the desynchronization between the cavity soliton and the external pump pulse. The mean-field equation reads ( ) where 2  is the second-order dispersion; ( ) loss; L  is the pump-cavity phase detuning;  is the average Kerr coefficient; L is the roundtrip length; and p A is the pump field. The operator F accounts for the effects that can be easily applied in the frequency domain, including the frequency-dependent filtering loss, the group velocity dispersion and the pump-cavity desynchronization; i.e. (S29)

Fiber cavity stabilization
The detailed experimental setup is shown in Fig. S6. To maintain a stable pumpcavity frequency detuning for soliton generation, the laser frequency is first locked to the fiber cavity resonance by sending a probe light in the backward direction and using the Pound-Drever-Hall locking technique. An acousto-optic frequency shifter is then used to tune the frequency of the pump pulse in the forward direction. The polarizations of the probe and the pump are adjusted orthogonal to each other to minimize their mutual interference.

Intensity cross correlation for soliton characterization
The intensity cross correlation setup for measuring the soliton pulse shape is shown in Fig. S7. The synchronous sampling pulse is generated through a multiplestage spectral broadening and pulse compression procedure. A portion of the Gaussian pump pulse is first amplified to a peak power of ~200 W, and sent through a 50-m highly nonlinear fiber (HNLF). The spectrum is broadened to ~2 nm and the pulse is compressed by a spectral shaper. A nonlinear amplifying loop mirror (NALM) composed of 2-m HNLF and 0.5-m Erbium-doped fiber (LIEKKI Er-110) is used to improve the pulse quality and further broaden the spectrum to ~6 nm. After passing through another 50-m HNLF, the spectral bandwidth exceeds 100 nm. A second spectral shaper is used to select the spectrum within 1527-1567 nm and shape the pulse to a Gaussian function with a full-width-at-half-maximum of 0.2 ps. The peak power of the output pulse is ~130 W. The pulse is much narrower than the solitons to be measured, making it possible to capture the fine temporal features with a high resolution.   Figure S8 shows the full data of the soliton transition process in Fig. 3

Effect of pump-cavity desynchronization
It is found that the soliton transition process is affected by the desynchronization between the pump pulse and the cavity. Figure S9a shows multiple intracavity power traces measured for slightly different pump pulse repetition rates. The simulation results are shown in Fig. S9b. The measured filter transfer function shown in Fig. 2c of the main paper is employed in the simulation. The desynchronization parameter (i.e., relative difference between the pump pulse repetition rate and the cavity FSR) varies between 4 3 10 −  . The widest single-soliton step is achieved when the desynchronization is slightly negative (of order of 5 10 − ). When the desynchronization magnitude gets larger, the single-soliton region becomes narrower and may even disappear. For large desynchronization, the intracavity field drops to the lower-branch homogenous state before reaching the single-soliton state. Simulations show that the soliton transition behavior may also be affected by the pump pulse shape as well as the asymmetry of the filter function.  Figure Figure S11 shows the soliton spectra measured before and after the spectral shaper.

Spectral evolution in one roundtrip
The uniform insertion loss of the spectral shaper has been subtracted. No significant change can be observed. The flat pedestal in the spectrum afore spectral shaper, which is more than 20 dB lower than the soliton spectral intensity, is mainly attributed to the amplifier spontaneous emission noise.

Numerical simulation of dispersion-less soliton microcomb
In this section, we show numerical simulation results of filter-driven dispersionless solitons generated within on-chip integrated microresonators. Figure S12a shows two conceptual designs including ring and Fabry-Pérot structures. For the ring cavity, spectral control is integrated through a periodic Bragg boundary that is implemented on the inner side of the ring waveguide. Therefore, the frequency components falling within the stopband can be confined in the cavity while the out-of-band frequencies experience high loss and will leak out. For the Fabry-Pérot cavity, the same goal can be accomplished by tailoring the two Bragg reflectors. Moreover, the photonic crystal structure provides a new degree of freedom for dispersion engineering to achieve negligible dispersion S5 .
The simulation is performed based on the following equation To obtain results more compatible with experiments, the third-order dispersion and Raman effect are also included in the model. The typical parameters of silicon nitride microresonators are employed and summarized in Table 1.
The comb line spacing ( c R ) is 50 GHz corresponding to a cavity roundtrip length ( L ) of 3 mm. The pump field is generated by modulating the phase and amplitude of a continuous-wave laser, given by  from the blue-detuned side to the red-detuned side as in usual experiments S8 ). The fullwidth-at-half-maximum is 74 fs. The comb spectrum is shown in Fig. S12c. There are ~250 lines spanning over 100 nm, within a 5-dB intensity range excluding the pump.
The tilt of spectral envelop towards longer wavelength is attributed to Raman induced self-frequency shift S9 .  Femtosecond soliton pulse. c Comb spectrum.